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?:$:i ~$:;;;""!It"j! .. i ChapterIII r ( Plasmas j r ,J 1. INTRODUCTION The discussion in Chapter II was concerned primarily with the behavior and properties of individual particles in a partially ionized gas. We wish now to turn our attention to the macroscopic behavior of collections of chargedparticles.Theseconsiderationswill lead to the introduction of two related fundamental parametersassociatedwith the electrical properties of a partially ionized gas, namely, the Debye length and the plasma frequency. As noted in Sec. II 8, the collective behavior of neighboring charged particles during a collision between two charged particles plays an essentialrole in section. the calculation of the particle momentum collision cross The notion ofcharged shielding involved here alsotransfer enters the description of the ionized gas region, called a sheath, immediately adjacent to a solid surface. The last three sections of this chapter are concerned with several topics which involve applications of the fundamental concepts introduced earlier. We discuss first the classical theory of electrostatic probes and their use in making measurementsof the properties of low-pressure ionized gases.We then discuss some of the concepts involved in the description of collision- i r '; r 1 !' ~ ( ; 'j J dominated ionized gasesadjacent to solid surfaces.Finally, we discuss the elementary theory of the propagation of electromagnetic radiation through an ionized gas and how diagnostic information about ionized gasescan be inferred from experiments which employ electromagnetic waves. 2. ELECTRICAL NEUTRALITY-THE DEBYE LENGTH A basic property of a partially ionized gas is its tendency towards electrical neutrality. If over a macroscopic volume the magnitudes of the charge) densities of the negative and positive particles differed just slightly, very ,.: largeelectrostatic forceswouldexist,for whichthepotentialenergyper f 126 : Section 2 .' Electrical Neutrality-The Debye Length 127 1 particle would enormously exceed the mean thermal energy. Unless very special mechanisms were involved to support such large potentials, the charged particles would move rapidly in such a way as to reduce these ~ potential differencesand therebyrestoreelectricalneutrality. i i I \ \ To obtain a quantitative estimate of the dimensions over which deviations from charge neutrality may occur, let us consider the following simplified model.. Let us sup~ose the ga~ is initia~ly .electrically neutral and that the electrons and Ions are unIformly distributed throughout space, as I indicatedschematicallyin Fig. l(a). Initially, the electronand ion number \ \ , I ( I + + + + + + + + ++ +:+ ,- -+ -+ -+ -+ - - - - - - - -- -- ~~~~~~~~ -+ -+ -+ -+ -+ -+ -+ -+ ~~+I~~~~~++ +1+ -1- -+ -+ -+ -+ - + + 0 + + + + + + y ++ -- +1++ 0 Yo 1 , + + + d Y i ! Ey Ey " I Ey(Yo) \, 1 ~ 0 0 (0) Initial condition Yo (b) Intermediatecondition Figure 1. Work necessaryto create a region of net positive charge. ~ densitieshave the common value n = ne = nj' We wish to calculate the work ~ , ~ necessary to displace the electrons bodily to the right through some distance, d. This will then be ,the work necessaryto create a region of net positive charge density pc = ne between the planes y = 0 and y = d. Let us supposethat Yo representssome intermediate displacement of the electrons, as shown in Fig. 1(b). The electric field E set up by a distribution of charge density pc is determined quite generally by Gauss' equation i C VoE=.e.-. (2.1) £0 For the one dimensional distribution in Fig. l(b), we have " 1 ~ dE" y::;O,y=O=E,,=O, y ( 2.2a) i 128 Plasmas Chapter dE" ne O~y~yo,-=-=>E,,=-y, dy £0 ne III (2.2b) £0 r Yo ~ y, -dE" d = 0 => = E" Yo = -ne ( E" ) y (2 Yo. . 2c) £0 1 J ' j the results expressed by equations (2.2), we have assumed In obtaining that the electric . contInuous. field is zero prior to the displacement and that E" is ~ " -I In the region to the right of Yo, the electric field acting on each electron is E,,(yo).The work necessaryto move each electron an additional distance dyo is dW = eE,,(yo)dyo. J Therefore, the total work that must be done on each electron to produce a total charge separation of distance d is ~ fd ne2 d2 W = eE,,(yo) dyo= - _ : 2. 0 (2.3) j £0 In particular, if this energy is to be derived from the mean thermal " = ADis " energy in the y-direction kT /2, then the corresponding distance d called the Debye length and is given by ne2Af, kT --=-, £0 2 2 :1 ! .J or r I [ ! j (2.4a) J( ! I In MKS units, AD = 69.0 ( T ) 1/2 n m. ~ " ) (2.4b); As an example,for the characteristic conditions in an MHO generator T = 2500oK and n = 1020m-3, we have AD~ 3.4 X 10-7 m. This distance may be compared to the value of the electron mean free path Ie ~ 1.3X 10-6 m, for the conditions specified in Exercise II 8.2. In this case, therefore, the Oebye length is about a factor of five less than the electron mean free path. Values of the Oebye length for various other conditions of interest j are shown in Fig. 12 of Chapter II. T 129 In 1929,Langmuir (1961)introduced the term plasmafor a partially ionized " gas in which AD is small compared to other macroscopic lengths of importance (for example, the macroscopic scale of change in electron number 1 density). Under such circumstances one may make the assumption of electrical neutrality, i.e.,nj ~ ne.The word plasma derivesfrom a Greek word . meaning"to mold" and was suggested to Langmuir by his observationsof the manner in which the positive column of a glow discharge tended to mold itself to the containing tube. r. Exercise 2.1. Consider a neutral plasma of charged particle number densityne= nj = 1014cm-3 and temperature2500oK: \. : !~ \' Ii \ 1. Supposethat in some manner all the electrons present in a sphere of radium 1 mm were suddenly removed. Calculate the resulting electric field (in voltsjm) at the sphere'ssurface. 2. Calculate the potential difference through which an electron would need to be accelerated in order to acquire a kinetic energy corresponding to the mean thermal energy of an electron in this plasma. \ ( 3. What is the maximum fraction of electrons that can be removed from the sphere such that the resulting potential difference between the center and \ I \ the surface of the sphere shall not exceedthe potential difference calculated in part 2? 4. Calculate the Debye length for this plasma. 3. SHEATHS l \ ( ; ! One of the most important situations in which charge neutrality does not prevail is in the region of a partially ionized gas immediately adjacent to a solid surface. Such regions are referred to as sheaths. The relevant macroscopicscalehere being the distancefrom the surface,we may anticipate that the assumption of charge neutrality will be violated in a region whose extent is of the order of AD' The detailed structure of sheaths may be quite varied, depending on many factors. The discussion of this section will be limited to the simplest of models and is aimed at bringing out the salient features of sheaths most expeditiously. An important aspect of the sheath problem concerns the disposition of charged particles which strike the solid surface. For many situations where the surface is cooled, it is possible to regard the surface as nonemitting and catalytic. Under such conditions the incident charged particles are either retained on the solid surfaceor they recombine and are returned to the gas as neutral particles. Let us consider the special caseof a floating electrode suddenly immersed into a stationaryplasma.We shall assume,for simplicity,that the electron